Antiderivatives
and
uses of derivatives and
antiderivatives
Ann Newsome
Definition of Antiderivative: Let f be a function of x. If F
is a function such that F’(x) = f(x), then F is an
antiderivative of f.
Ex. If F(x) = x2
and f (x) = 2x
then F ¢(x) = f (x)
so F(x) is an antiderivative of f.
Antiderivatives are not unique.
If F(x) is an antiderivative of f, and C is any constant,
then F(x) + C is also an antiderivative of f.
Ex.
f(x) = x 2
1 3
F(x) = x
3
is an antiderivative
1 3
F(x) = x +1
3
is also an antiderivative
1 3
F(x) = x + 2
3
is also an antiderivative
1 3
F(x) = x + C
3
defines all antiderivatives of f
Power Rule for Antiderivatives:
1
k+1
If f(x) = x , k ¹ -1, then F(x) =
×x +C
k +1
k
If k were -1, the denominator would be zero.
To verify this formula:
x k+1
F(x) =
+C
k +1
(k +1)x k
F ¢(x) =
+0
k +1
= x k verifying the formula.
Ex.
f(x) = x 5 + 3x 2 + 4
1 6 3 3
F(x) = x + x + 4x + C
6
3
1 6 3
F(x) = x + x + 4x + C
6
Confirm by taking the derivative:
f(x) = x 5 + 3x 2 + 4
✔
Using your graphing calculator, examine the graph of
1
-1
y = = x for x > 0
x
1
-1
f ¢(x) = = x x > 0
x
If this graph is the derivative of f(x),
what do we know about f ?
Q. Is f increasing or decreasing?
A. We know f is increasing
because its derivative is positive.
Q. What is the concavity of f ?
A. We know f is concave
down because f’ is decreasing.
Can you think of a function that is always increasing, always
concave down, and has a domain (0,∞)?
Hint: It’s not a polynomial.
y = lnx has the right characteristics.
Explore this possibility using the graphing calculator.
Fact: The antiderivative of f(x) =
is F(x) = lnx +C.
1
x
Ex. What function has f(x) = 4x 3 as its derivative?
F(x) =
4 3+1
4
x =x
3+1
Are there any other possible antiderivatives?
Yes, F(x) = x 4 + 2 is an example.
Ex. What function has f(x) = x 6 - 5x 2 as its derivative?
1
5
F(x) = x 7 x 2+1 + C
(2 + 1)
7
1
5
= x7 - x3 + C
7
3
f(x) = x6 - 5x2
✔
Confirm the results by
taking the derivative.
Last problem: p. 121, #44:
Q. For which values of x does the slope of the line tangent to the
curve f(x) = -x 3 + 3x 2 +1 take on its largest value?
A. To find the slope I will take the derivative of the function.
f ¢(x) = -3x 2 +6x I now need to find where this function has a
maximum value. To find this I will look at the
derivative of f’.
f ¢¢(x) = -6x +6
f ¢¢(x) = -6x + 6 = 0
x =1
Where f’ has a maximum value, f” will have
a zero.
f ¢¢
+
1
−
f’ has a local max at x = 1, where f” changes from
positive to negative. f’ increases before 1 and
decreases afterwards, so the greatest slope is at x = 1.